Perturbative analysis of disordered Ising models close to criticality

TL;DR

This paper analyzes a disordered 2D Ising model near criticality, demonstrating that with low probability of supercritical couplings, a convergent cluster expansion exists, ensuring the free energy is infinitely differentiable but not analytic.

Contribution

It introduces a novel application of graded cluster expansions and stochastic domination to disordered Ising models near critical points.

Findings

Cluster expansion converges with high probability under certain conditions.

Free energy is infinitely differentiable but not analytic.

Provides a framework for analyzing disordered systems near criticality.

Abstract

We consider a two-dimensional Ising model with random i.i.d. nearest-neighbor ferromagnetic couplings and no external magnetic field. We show that, if the probability of supercritical couplings is small enough, the system admits a convergent cluster expansion with probability one. The associated polymers are defined on a sequence of increasing scales; in particular the convergence of the above expansion implies the infinite differentiability of the free energy but not its analyticity. The basic tools in the proof are a general theory of graded cluster expansions and a stochastic domination of the disorder.